B-spline

Remarks

$B_{i,p}$ is a spline of degree $p$ with knots $t_i, \dots, t_{i+p+1}$; if $t_i < t_{i+p+1}$, it is positive on $(t_i, t_{i+p+1})$ and zero outside $[t_i, t_{i+p+1})$. On $[t_p, t_{m-p})$ the B-splines of degree $p$ sum to $1$, so they form a partition of unity there. At a knot of multiplicity $r \le p$, the B-splines of degree $p$ have $p - r$ continuous derivatives. An equivalent definition writes $B_{i,p}$ as $(t_{i+p+1} - t_i)$ times a divided difference of the truncated power $(\cdot - x)_+^p$ at $t_i, \dots, t_{i+p+1}$.

Example: knots $0, 1, 2$ and $p = 1$. Then $B_{0,1}(x) = x \, B_{0,0}(x) + (2 - x) \, B_{1,0}(x)$, which is $x$ on $[0, 1)$, $2 - x$ on $[1, 2)$ and $0$ elsewhere: the hat function.

Non-example: let $g$ be $x$ on $[0, 1)$, $1$ on $[1, 2)$ and $0$ elsewhere. It is piecewise linear on the same knots, but it is not $B_{0,1}$. It differs from the hat function on $(1, 2)$, and it jumps at $2$.